Exam 7: Systems Of Equations and Inequalities

An investor has $450,000 to invest in two types of investments.Type A pays 6% annually and type B pays 7% annually.To have a well-balanced portfolio, the investor imposes the following conditions.At least one-third of the total portfolio is to be allocated to type A investments and at least one-third of the portfolio is to be allocated to type B investments.What is the optimal amount that should be invested in each investment?

Write the partial fraction decomposition of the improper rational expression. $\frac { x ^ { 2 } + 4 x } { x ^ { 2 } + x + 1 }$

Use any method to solve the system of linear equations, find (x,y). $\left\{ \begin{array} { c } 4 x - 5 y = 13 \\5 x + y = 9\end{array} \right.$

Use back-substitution to solve the system of linear equations. $\left\{ \begin{array} { c l } 2 x - y + 5 z & = 24 \\y + 2 z & = 9 \\z & = 7\end{array} \right.$

Solve the system by the method of substitution. $\left\{ \begin{aligned}\frac { 1 } { 5 } x + \frac { 1 } { 2 } y & = 8 \\x + y & = 21\end{aligned} \right.$

Determine whether the ordered triple is a solution of the system of equations. $\left\{ \begin{array} { l } 3 x + 4 y - z = 16 \\5 x - y + 2 z = 26 \\2 x - 3 y + 7 z = 30\end{array} \right.$ $( 4,2,4 )$

Use back-substitution to solve the system of linear equations. $\left\{ \begin{array} { c c } x - y + 2 z & = 22 \\3 y - 8 z & = - 9 \\z & = - 9\end{array} \right.$

For the rational expression $\overline { ( x + 60 ) ( x - 60 ) ^ { 2 } }$ the partial fraction decomposition is of the form $\frac { A } { x + 60 } + \frac { B } { ( x - 60 ) ^ { 2 } }$ .

Find the equilibrium point (x,p) of the demand and supply equations.(The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations.) Demand Supply p=570-0.5x p=370+0.3x

Find the maximum value of the objective function and where it occurs, subject to the indicated constraints. ​ Objective function: ​ Z = 4x + 5y ​ Constraints: ​ X ≥ 0 ​3x - y ≤ 9 2x + 3y ≥ 6 X + 4y ≤ 16 ​​ ​

A chemist needs 10 liters of a 25% acid solution.The solution is to be mixed from three solutions whose concentrations are 10%, 20%, and 50%.How many liters of each solution will satisfy each condition? Use 2 liters of the 50% solution. ​

Solve the system by the method of substitution. $\left\{ \begin{array} { l } 0.5 x + 3.2 y = 5.3 \\0.3 x - 1.4 y = - 1.8\end{array} \right.$

Use any method to solve the system of linear equations, find (x,y). $\left\{ \begin{array} { c } 4 x - 3 y = 16 \\- 5 x + 3 y = - 23\end{array} \right.$

Solve the system of linear equations and check any solution algebraically. $\left\{ \begin{array} { l } 5 x + 3 y + 17 z = 0 \\6 x + 4 y + 22 z = 0 \\5 x + 2 y + 19 z = 0\end{array} \right.$

Select the correct graph of the inequality. $y \leq \frac { 5 } { 1 + x ^ { 2 } }$

Seven hundred gallons of 87-octane gasoline is obtained by mixing 87-octane gasoline with 90-octane gasoline.Use a graphing utility to graph the two given equations.Let x and y represent the number of gallons of the 87-octane gasoline and 90-octane gasoline.As the number of gallons of the 87-octane gasoline increases, how does the number of gallons of the 90-octane gasoline change $\left\{ \begin{array} { l } x + y = 700 \\87 x + 90 y = 60,900\end{array} \right.$

Select the region determined by the constraints.Then find the maximum value of the objective function (if possible) and where it occurs, subject to the indicated constraints. ​ Objective function: ​ Z = 8x + 9y ​ Constraints: ​ X ≥ 0 Y ≥ 0 X + y ≥ 8 3x + 5y ≥ 30 ​

Find the consumer surplus and producer surplus. Demand $p = 100 - 0.00006 x$ Supply $p = 90 + 0.00004 x$

Write the partial fraction decomposition of the rational expression.Check your result algebraically. $\frac { 5 x ^ { 2 } + 3 x - 1 } { x ^ { 2 } ( x + 1 ) }$

The linear programming problem has an unusual characteristic.Select a graph of the solution region for the problem and describe the unusual characteristic.Find the minimum value of the objective function (if possible) and where it occurs. ​ Z = x + 3y ​ Constraints: ​ X ≥ 0 Y ≥ 0 X + 2y ≤ 4 2x + y ≤ 4 ​